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Computers and Mathematics with Applications, ISSN 0898-1221, 05/2019, Volume 77, Issue 10, pp. 2725 - 2739
In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional -Laplacian where denotes... 
Multiple solutions | Schrödinger–Kirchhoff problems | Fractional [formula omitted]-Laplacian system | Sign-changing potential | Fractional p-Laplacian system | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | EQUATIONS | Schrodinger-Kirchhoff problems | GROUND-STATES
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 07/2015, Volume 121, pp. 370 - 381
In this paper, we prove the following result. Let be any real number between and . Assume that is a solution of for some and . Then must be constant throughout... 
[formula omitted]-harmonic functions | Fourier analysis | Liouville theorem | Poisson representations | The fractional Laplacian | α-harmonic functions | INTEGRAL-EQUATION | MATHEMATICS | MATHEMATICS, APPLIED | OPERATOR | alpha-harmonic functions | EXTENSION | FORMULA | Theorems | Real numbers | Texts | Potential theory | Nonlinearity | Constants
Journal Article
Numerical Functional Analysis and Optimization, ISSN 0163-0563, 06/2017, Volume 38, Issue 6, pp. 738 - 753
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 04/2012, Volume 75, Issue 6, pp. 3210 - 3217
In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional -Laplacian equation where , is a Caputo... 
Fractional differential equation | [formula omitted]-Laplacian operator | Resonance | Boundary value problem | Existence | Coincidence degree | p-Laplacian operator | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | CALCULUS APPROACH | Differential equations
Journal Article
Frontiers of Mathematics in China, ISSN 1673-3452, 4/2017, Volume 12, Issue 2, pp. 389 - 402
We investigate the nonnegative solutions of the system involving the fractional Laplacian: $$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta... 
method of moving planes | 35A05 | Kelvin transform | Liouville theorem | 58J70 | 47G30 | Mathematics, general | Mathematics | Fractional Laplacian | 35J45 | 35S05 | MATHEMATICS | R-N | POSITIVE SOLUTIONS | CLASSIFICATION | DIFFUSION | ELLIPTIC-EQUATIONS | DOMAINS | Studies | Laplace transforms | Topological manifolds | Mathematical analysis | Symmetry | Functions (mathematics) | Planes | Independent variables | Texts | Formulas (mathematics)
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 05/2019, Volume 182, pp. 248 - 262
This paper deals with the following Choquard equations with fractional -Laplacian where , , and . By constructing a decay at infinity theorem and a narrow... 
Direct method of moving planes | Fractional [formula omitted]-Laplacian | Radial symmetry | Choquard equations | Fractional p-Laplacian | MATHEMATICS | MATHEMATICS, APPLIED | MAXIMUM-PRINCIPLES | ELLIPTIC PROBLEM | Mathematical analysis | Symmetry
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 07/2017, Volume 158, pp. 109 - 131
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 08/2013, Volume 87, pp. 116 - 125
In this paper, we study the existence of ground state solutions for the following critical fractional Laplacian equation where is a bounded domain. We will... 
Ground state solution | Fractional Laplacian equation | [formula omitted] condition | (PS) condition | OBSTACLE PROBLEM | MATHEMATICS | MATHEMATICS, APPLIED | OPERATOR | REGULARITY | EXPONENT | BOUNDARY | Mathematical analysis | Nonlinearity | Ground state
Journal Article
Numerical Functional Analysis and Optimization, ISSN 0163-0563, 08/2019, Volume 40, Issue 11, pp. 1315 - 1343
Journal Article
Communications in Partial Differential Equations, ISSN 0360-5302, 06/2015, Volume 40, Issue 6, pp. 1070 - 1095
We study elliptic gradient systems with fractional laplacian operators on the whole space where u: R n  → R m , H ∈ C 2, γ (R m ) for γ > max (0, 1 − 2 min {s... 
Elliptic systems | Liouville theorems | Symmetry of solutions | De Giorgi's conjecture | 35B53 | Fractional Laplacian | 35J47 | 34A08 | 35B08 | MATHEMATICS, APPLIED | INEQUALITY | EQUATIONS | ASYMPTOTIC-BEHAVIOR | CONJECTURE | LAPLACIAN | MATHEMATICS | REGULARITY | GIORGI | THEOREMS | Partial differential equations | Symmetry | Operators | Liouville theorem | Images | Estimates | Optimization
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 2011, Volume 74, Issue 5, pp. 1926 - 1936
Journal Article
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, ISSN 0360-5302, 08/2019, Volume 44, Issue 8, pp. 637 - 680
We consider some nonlinear fractional Schrodinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or... 
variational methods | MATHEMATICS | NLS EQUATIONS | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | STATES | POSITIVE SOLUTIONS | LIMIT | magnetic fractional Laplacian | OPERATORS | Fractional magnetic Kato's inequality | Nonlinear equations | Minimax technique | Schroedinger equation | Magnetic fields | Mathematical analysis
Journal Article
Computers and Mathematics with Applications, ISSN 0898-1221, 10/2019, Volume 78, Issue 8, pp. 2593 - 2617
In this paper we deal with the following fractional Choquard equation where is a small parameter, , , , is the fractional -Laplacian, is a positive continuous... 
Variational methods | Fractional Choquard equation | Fractional [formula omitted]-Laplacian | Minimax technique | Continuity (mathematics)
Journal Article
Complex Variables and Elliptic Equations, ISSN 1747-6933, 03/2018, Volume 63, Issue 3, pp. 346 - 359
In this paper, we use variational methods to study existence of solutions for the following fractional p-Laplacian equations of Schrödinger-Kirchhoff type... 
variational methods | fractional Schrödinger equation | mountain pass theorem | 35A15 | 47G20 | 35J60 | Fractional p-Laplacian | 35R11 | Fractional schrödinger equation | Mountain pass theorem | Variational methods | fractional Schrodinger equation | MATHEMATICS | Operators (mathematics) | Mathematical analysis | Formulas (mathematics)
Journal Article
Stochastic Processes and their Applications, ISSN 0304-4149, 07/2014, Volume 124, Issue 7, pp. 2479 - 2516
Suppose that and . Let be such that each is a signed measure on belonging to the Kato class . In this paper, we consider the stochastic differential equation... 
Transition density | Symmetric [formula omitted]-stable process | Lévy system | Exit time | Kato class | Heat kernel | Gradient operator | Green function | Boundary Harnack inequality | Symmetric α-stable process | BROWNIAN-MOTION | Levy system | STATISTICS & PROBABILITY | Symmetric alpha-stable process | FRACTIONAL LAPLACIAN | OPERATORS
Journal Article
Communications in Partial Differential Equations, ISSN 0360-5302, 11/2019, Volume 44, Issue 11, pp. 1100 - 1139
In this article, we study the logarithmic Laplacian operator which is a singular integral operator with symbol We show that this operator has the integral... 
Boundary decay | logarithmic symbol | fractional Laplacian | 35R11 | 35D30 | 35B51 | 35B50 | maximum principle | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | INEQUALITY | NONLOCAL OPERATORS | Operators (mathematics) | Domains | Mathematical analysis | Integrals | Eigenvalues | Maximum principle | Dirichlet problem | Eigenvectors | Gamma function
Journal Article
Applied Mathematics Letters, ISSN 0893-9659, 09/2014, Volume 35, Issue 1, pp. 1 - 6
In this paper, we prove the global regularity to the Cauchy problem of 3D incompressible magnetohydrodynamic- (MHD- ) model with fractional diffusion. We prove... 
Fractional Laplacian | Incompressible 3D MHD-[formula omitted] model | Global regularity | Incompressible 3D MHD-α model | Incompressible 3D MHD-alpha model | CRITERIA | SYSTEM | DISSIPATION | MATHEMATICS, APPLIED | FLUID | 2D MHD EQUATIONS | EULER
Journal Article
Complex Variables and Elliptic Equations, ISSN 1747-6933, 07/2017, Volume 62, Issue 7, pp. 1002 - 1014
In this paper, narrow region principle for cooperative systems involving fractional Laplacian is developed. As an application, some Liouville type results of... 
35J60 | 53C21 | moving planes | radial symmetry | non-existence | 58J05 | Fractional Lane-Emden system | narrow region principle | Fractional Lane–Emden system | POSITIVE SOLUTIONS | INTEGRAL-EQUATIONS | CLASSIFICATION | MONOTONICITY | LAPLACIAN | MATHEMATICS | NONLINEAR ELLIPTIC-EQUATIONS | MOVING SPHERES
Journal Article
Communications in Partial Differential Equations, ISSN 0360-5302, 01/2019, Volume 44, Issue 1, pp. 20 - 50
We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on , including the case of the critical... 
Primary 47D08 | 46E35 | heat kernel | Hardy inequality | Fractional Laplacian | Secondary 31C05 | 60J35 | MATHEMATICS | MATHEMATICS, APPLIED | UPPER-BOUNDS | INEQUALITIES | HEAT-EQUATION | ELLIPTIC-OPERATORS | SCHRODINGER-OPERATORS | KERNELS
Journal Article
by Chen, CS and Bao, JF
BOUNDARY VALUE PROBLEMS, ISSN 1687-2770, 08/2016, Volume 2016, Issue 1, pp. 1 - 16
In this paper, we study the existence, nonexistence, and multiplicity of solutions to the following fractional -Laplacian equation: (-Delta)(p)(s) u +... 
variational methods | MATHEMATICS | MATHEMATICS, APPLIED | NONTRIVIAL SOLUTION | (PS)(c) condition | Q)-LAPLACIAN | fractional p&q-Laplacian equation | Operators | Boundary value problems | Parameters | Variational methods | Mathematical analysis | Texts | Nonlinearity | Formulas (mathematics)
Journal Article
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